## TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. II: General framework.(English)Zbl 0662.65083

[For part I see the authors, The Runge-Kutta local projection $$P^ 1$$- discontinuous-Galerkin finite element method for scalar conservation laws, IMA Preprint Series # 388, University of Minesota (1988).]
This is the second paper in a series in which we construct and analyze a class of TVB (total version bounded) discontinuous Galerkin finite element methods for solving conservation laws $$u_ t+\sum^{d}_{i=1}(f_ i(u))_{x_ i}=0$$. In this paper we present a general framework of the methods, up to any order of formal accuracy, using scalar one-dimensional initial value and initial-boundary problems as models. In these cases we prove TVBM (total variation bounded in the means), TVB, and convergence of the schemes. Numerical results using these methods are also given. Extensions to systems and/or higher dimensions will appear in future papers.

### MSC:

 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35L65 Hyperbolic conservation laws
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